Composition operators on the polydisc

TL;DR

This paper investigates the boundedness of composition operators on weighted Bergman and Hardy spaces over the polydisc, providing new criteria and the rank sufficiency theorem for these operators.

Contribution

It introduces the rank sufficiency theorem for arbitrary polydiscs and offers a simple boundedness criterion for composition operators on the bidisc and tridisc spaces.

Findings

Established the rank sufficiency theorem for arbitrary polydiscs.

Provided a simple criterion for boundedness on the bidisc.

Achieved a consistent characterization for the tridisc case.

Abstract

We study the boundedness of composition operators on the weighted Bergman spaces and the Hardy space over the polydisc. For arbitrary polydisc we prove the rank sufficiency theorem which, in particular, provides us with a simple criterion describing boundedness of composition operators on the spaces over the bidisc. Such a consistent characterization is obtained for the classical Bergman space over the tridisc.